Sistemas y Señales Biomédicos

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-01-20

Sistemas y Señales Biomedicos - SYSB

Signal Classification

Signal Classification – Bounded

Signal Classification – Compact Support

Signal Classification – Causal

Signal Classification - Even/Odd

Even

\[f\left(t\right) = f\left(-t\right)\] \[f\left[t\right] = f\left[-t\right]\]

Odd

\[f\left(t\right) = -f\left(-t\right)\] \[f\left[t\right] = -f\left[-t\right]\]

Signal Classification

Decomposition

All signal can be decomposed in two signals: one even, one odd.

\[x(t) = x_{even}(t) + x_{odd}(t)\]

Where:

\[x_{even}(t) = \frac{x(t)+x(-t)}{2} \] \[x_{odd}(t) = \frac{x(t)-x(-t)}{2} \]

Example

Example

Decompose the signal \(x(t)=e^{t}\) into its even and odd parts

Example

\[x_{\text{even}}(t) = \frac{x(t) + x(-t)}{2}\]

\[x_{\text{odd}}(t) = \frac{x(t) - x(-t)}{2}\]

\[x(-t) = e^{-t}\]

\[x_{\text{even}}(t) = \frac{e^t + e^{-t}}{2} = \cosh(t)\]

\[x_{\text{odd}}(t) = \frac{e^t - e^{-t}}{2} = \sinh(t)\]

\[x(t) = x_{\text{even}}(t) + x_{\text{odd}}(t)\]

\[e^t = \cosh(t) + \sinh(t)\] ​

Example

Signal Transformations

Types of Transformations

Signals can undergo two types of transformations:

  1. Independent variable transformations (affect the time or input axis).
  2. Dependent variable transformations (affect the amplitude or output axis).

Independent Variable Transformations

Time Scaling

  • Definition: Changes the time scale of the signal. [ x(at), a > 1 , < a < 1 ]
  • Example: If ( x(t) = (t) ), then ( x(2t) ) is compressed.

Time Shifting

  • Definition: Shifts the signal in time. [ x(t - t_0) ]
  • Example: ( x(t - 2) ) shifts the signal 2 units to the right.

Time Reversal

  • Definition: Flips the signal across the vertical axis. [ x(-t) ]
  • Example: If ( x(t) = t^2 ), then ( x(-t) = t^2 ) (even signal).

Dependent Variable Transformations

Amplitude Scaling

  • Definition: Multiplies the amplitude by a scalar factor. [ a x(t), a > 1 , < a < 1 ]
  • Example: If ( x(t) = (t) ), then ( 2x(t) ) doubles the amplitude.

Amplitude Shifting

  • Definition: Adds a constant value to the amplitude. [ x(t) + c ]
  • Example: If ( x(t) = (t) ), then ( x(t) + 2 ) shifts the signal up by 2 units.

Combined Transformations

Example

Consider: [ y(t) = 2 x(3t - 1) + 1 ] 1. Time compression: ( x(3t) ) compresses the signal. 2. Time shift: ( x(3t - 1) ) shifts it to the right by 1 unit. 3. Amplitude scaling: ( 2 x(3t - 1) ) amplifies the signal. 4. Amplitude shift: ( +1 ) shifts it upward.

Visualization Example in Python